Sub-Doppler laser cooling using electromagnetically induced transparency

We propose a sub-Doppler laser-cooling mechanism that takes advantage of the unique spectral features and extreme dispersion generated by the phenomenon of electromagnetically induced transparency (EIT). EIT is a destructive quantum interference phenomenon experienced by atoms with multiple internal quantum states when illuminated by laser fields with appropriate frequencies. By detuning the lasers slightly from the “dark resonance,” we observe that, within the transparency window, atoms can be subject to a strong viscous force, while being only slightly heated by the diffusion caused by spontaneous photon scattering. In contrast to other laser-cooling schemes, such as polarization gradient cooling or EIT-sideband cooling, no external magnetic field or strong external confining potential is required. Using a semiclassical approximation, we derive analytically quantitative expressions for the steady-state temperature, which is confirmed by full quantum mechanical numerical simulations. We find that the lowest achievable temperatures approach the single-photon recoil energy. In addition to dissipative forces, the atoms are subject to a stationary conservative potential, leading to the possibility of spatial confinement. We find that under typical experimental parameters, this effect is weak and stable trapping is not possible.

Figure 1

The system we consider consists of atoms transversely confined so that they are able to move only in the $x$ direction. The probe lasers consist of a pair of counterpropagating beams aligned with the $x$ axis with equal intensity as characterized by the Rabi frequency ${\mathrm{\Omega }}_p$. The coupling laser propagates perpendicularly to the probe beams as shown and has intensity characterized by the Rabi frequency ${\mathrm{\Omega }}_c$.

Figure 2

(a) Lambda-type three-level atoms driven by two laser beams. (b) The real (solid line) and imaginary (dashed line) parts of the susceptibility $\chi$ are shown as a function of detuning ${\mathrm{\Delta }}_p$ for the case ${\mathrm{\Delta }}_c=0$. The $\text{Re}\left[\chi \right]$ characterizes refraction of the probe light, while the $\text{Im}\left[\chi \right]$ characterizes absorption. The inset magnifies the small detuning region to highlight the behavior of the susceptibility in the transparency window. The parameters used for the plot are ${\mathrm{\Omega }}_p=10E_r/\hslash , {\mathrm{\Omega }}_c=400E_r/\hslash$, and ${\gamma }_3=2000E_r/\hslash$.

Figure 3

Friction forces as a function of atom velocity. The two radiation forces (dashed lines) exerted by the two counterpropagating beams give rise to the net friction force (solid line). For blue-detuned probe beams, atoms with positive velocities experience a negative force and vice versa, corresponding to cooling. The magnitude of the friction force is maximized for atoms traveling at the critical velocity $\pm v_E$ shown, which determines the capture range. The parameters used were $kx=0, {\mathrm{\Delta }}_p=40E_r/\hslash , {\mathrm{\Omega }}_p=20E_r/\hslash , {\mathrm{\Omega }}_c=400E_r/\hslash$, and ${\gamma }_3=2000E_r/\hslash$.

Figure 4

Friction coefficient $\eta$ (solid line) and diffusion coefficient $D$ (dashed line) as a function of detuning. Both the friction and diffusion coefficients are zero at ${\mathrm{\Delta }}_p=0$. The friction $\eta$ reaches a maximum at approximately ${\mathrm{\Delta }}_p=2{\mathrm{\Omega }}_c^2/{\gamma }_3$, while the diffusion $D$ is maximized at approximately ${\mathrm{\Delta }}_p={\mathrm{\Omega }}_c$. The parameters used were ${\mathrm{\Omega }}_p=20E_r/\hslash , {\mathrm{\Omega }}_c=400E_r/\hslash$, and ${\gamma }_3\approx {\gamma }_1=2000E_r/\hslash$.

Figure 5

Final temperature reached as a function of the detuning. The parameters were $E_r/\hslash =5\mathrm{kHz}, {\mathrm{\Omega }}_c=2\mathrm{MHz}=400E_r/\hslash , {\mathrm{\Omega }}_p=100\mathrm{kHz}=20E_r/\hslash , {\gamma }_3\approx {\gamma }_1=10\mathrm{MHz}=2000E_r/\hslash$. The truncated momentum basis ranged from $-50\hslash k$ to $50\hslash k$. The analytical formula (solid line), semiclassical numerical results (triangle), and MCWF numerical results (circle) are compared. They agree well when the detuning is significantly larger than the recoil energy, but discrepancies appear when the detuning is near the recoil energy, as shown in the inset plot. The standard errors for the small detuning points are of the order of 0.1 as shown by the error bars in the inset, while the standard errors for large detunings points are of the order of 0.01, which are too small to be depicted in the large plot.

Figure 6

(a) The friction coefficient as a function of $2{\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c$, with the parameters ${\gamma }_3\approx {\gamma }_1=2000E_r/\hslash$ and ${\mathrm{\Omega }}_c=400E_r/\hslash$ at three detunings within the absorption dip; they are ${\mathrm{\Delta }}_p=10E_r/\hslash$ (round), ${\mathrm{\Delta }}_p=30E_r/\hslash$ (square), and ${\mathrm{\Delta }}_p=50E_r/\hslash$ (diamond), respectively. (b) The ratio between the lattice depth and the final kinetic energy as a function of $2{\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c$, with the same simulation parameters as (a). (c) Five trajectories, representing the atom position $x$ as a function of time. The simulation parameters are extracted from the circle in (a). At about $t\sim 1/\eta$, as shown by the dashed line, the atom is cooled down. After short periods of weak trapping within the wells of the lattice, the atom may gain sufficient energy to propagate over multiple sites before being cooled and recaptured. (d) The position dependence of the density distribution, with the same parameters as used in (c). The histogram is obtained from the simulations by recording the position of each atom after reaching equilibrium. The solid line represents the density distribution that is analytically derived by assuming that the thermodynamics of the equilibrated atomic gas obeys the Boltzmann distribution.